We study observations of transmission pairs as shown in Figure REF . The first person gets infected at time point \(T_1\)  which is unobserved. However, it is known to lie in the interval \([0,W]\) . The exact conditional distribution of \(T_1\)  in \([0,W]\)  is allowed to also depend on a random variable \(C\)  which determines for example the location of the infection. In certain situations, it will turn out that the likelihood will depend only on \(\min (W,S_1)\) , where \(S_1\ge T_1\)  denotes the time at which Person 1 shows symptoms. In that sense, observation of \(W\)  is only then required if \(W\le S_1\) . Otherwise we do not have to observe \(W\)  but we have to observe that \(W\ge S_1\) . The incubation period is defined as \(I_1:=S_1-T_1\) . In an analogue fashion, \(S_2,T_2,I_2\)  are defined for the second person. Moreover, Person 1 is known to have infected Person 2. Person 2 shows symptoms at the observed time \(S_2\) . But, of course, the infection time \(T_2\)  is unobserved. The generation time is thus \(G:=T_2-T_1\)  and it is also unobserved, while the serial interval \(S:=S_2-S_1\)  can be observed. Such data was for example collected by [1]}, [2]}, [3]}.
<FIGURE>